tanhf (example 3.4)

Average Error: 31.0 → 0.7

Time: 6.9s

Precision: 64

Math

\[\frac{1 - \cos x}{\sin x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.014503396267940102:\\ \;\;\;\;\frac{\frac{1}{\sin x} \cdot \frac{1}{\sin x} - \frac{\cos x}{\sin x} \cdot \frac{\cos x}{\sin x}}{\frac{1}{\sin x} + \frac{\cos x}{\sin x}} \cdot 1\\ \mathbf{elif}\;x \le 0.0228016167359906367:\\ \;\;\;\;\left(\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x + \left(-1\right) \cdot \frac{\sin x}{\cos x}}{\left(-\sin x\right) \cdot \frac{\sin x}{\cos x}} \cdot 1\\ \end{array}\]

C

double code(double x) {
	return ((double) (((double) (1.0 - ((double) cos(x)))) / ((double) sin(x))));
}

↓

double code(double x) {
	double VAR;
	if ((x <= -0.014503396267940102)) {
		VAR = ((double) (((double) (((double) (((double) (((double) (1.0 / ((double) sin(x)))) * ((double) (1.0 / ((double) sin(x)))))) - ((double) (((double) (((double) cos(x)) / ((double) sin(x)))) * ((double) (((double) cos(x)) / ((double) sin(x)))))))) / ((double) (((double) (1.0 / ((double) sin(x)))) + ((double) (((double) cos(x)) / ((double) sin(x)))))))) * 1.0));
	} else {
		double VAR_1;
		if ((x <= 0.022801616735990637)) {
			VAR_1 = ((double) (((double) (((double) (0.041666666666666664 * ((double) pow(x, 3.0)))) + ((double) (((double) (0.004166666666666667 * ((double) pow(x, 5.0)))) + ((double) (0.5 * x)))))) * 1.0));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) sin(x)) + ((double) (((double) -(1.0)) * ((double) (((double) sin(x)) / ((double) cos(x)))))))) / ((double) (((double) -(((double) sin(x)))) * ((double) (((double) sin(x)) / ((double) cos(x)))))))) * 1.0));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Target

Original	31.0
Target	0.0
Herbie	0.7

\[\tan \left(\frac{x}{2}\right)\]

Derivation

1. Split input into 3 regimes

2. if x < -0.014503396267940102

   1. Initial program 0.9

      \[\frac{1 - \cos x}{\sin x}\]
   2. Using strategy rm

   3. Applied add-log-exp 1.1

      \[\leadsto \color{blue}{\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 1.1

      \[\leadsto \log \left(e^{\frac{1 - \cos x}{\color{blue}{1 \cdot \sin x}}}\right)\]

   6. Applied *-un-lft-identity 1.1

      \[\leadsto \log \left(e^{\frac{\color{blue}{1 \cdot \left(1 - \cos x\right)}}{1 \cdot \sin x}}\right)\]

   7. Applied times-frac 1.1

      \[\leadsto \log \left(e^{\color{blue}{\frac{1}{1} \cdot \frac{1 - \cos x}{\sin x}}}\right)\]

   8. Applied exp-prod 1.2

      \[\leadsto \log \color{blue}{\left({\left(e^{\frac{1}{1}}\right)}^{\left(\frac{1 - \cos x}{\sin x}\right)}\right)}\]

   9. Applied log-pow 0.9

      \[\leadsto \color{blue}{\frac{1 - \cos x}{\sin x} \cdot \log \left(e^{\frac{1}{1}}\right)}\]

   10. Simplified 0.9

       \[\leadsto \frac{1 - \cos x}{\sin x} \cdot \color{blue}{1}\]
   11. Using strategy rm

   12. Applied div-sub 1.2

       \[\leadsto \color{blue}{\left(\frac{1}{\sin x} - \frac{\cos x}{\sin x}\right)} \cdot 1\]
   13. Using strategy rm

   14. Applied flip-- 1.6

       \[\leadsto \color{blue}{\frac{\frac{1}{\sin x} \cdot \frac{1}{\sin x} - \frac{\cos x}{\sin x} \cdot \frac{\cos x}{\sin x}}{\frac{1}{\sin x} + \frac{\cos x}{\sin x}}} \cdot 1\]

   if -0.014503396267940102 < x < 0.022801616735990637

   1. Initial program 59.9

      \[\frac{1 - \cos x}{\sin x}\]
   2. Using strategy rm

   3. Applied add-log-exp 59.9

      \[\leadsto \color{blue}{\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 59.9

      \[\leadsto \log \left(e^{\frac{1 - \cos x}{\color{blue}{1 \cdot \sin x}}}\right)\]

   6. Applied *-un-lft-identity 59.9

      \[\leadsto \log \left(e^{\frac{\color{blue}{1 \cdot \left(1 - \cos x\right)}}{1 \cdot \sin x}}\right)\]

   7. Applied times-frac 59.9

      \[\leadsto \log \left(e^{\color{blue}{\frac{1}{1} \cdot \frac{1 - \cos x}{\sin x}}}\right)\]

   8. Applied exp-prod 59.9

      \[\leadsto \log \color{blue}{\left({\left(e^{\frac{1}{1}}\right)}^{\left(\frac{1 - \cos x}{\sin x}\right)}\right)}\]

   9. Applied log-pow 59.9

      \[\leadsto \color{blue}{\frac{1 - \cos x}{\sin x} \cdot \log \left(e^{\frac{1}{1}}\right)}\]

   10. Simplified 59.9

       \[\leadsto \frac{1 - \cos x}{\sin x} \cdot \color{blue}{1}\]

   11. Taylor expanded around 0 0.0

       \[\leadsto \color{blue}{\left(\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\right)} \cdot 1\]

   if 0.022801616735990637 < x

   1. Initial program 0.9

      \[\frac{1 - \cos x}{\sin x}\]
   2. Using strategy rm

   3. Applied add-log-exp 1.0

      \[\leadsto \color{blue}{\log \left(e^{\frac{1 - \cos x}{\sin x}}\right)}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 1.0

      \[\leadsto \log \left(e^{\frac{1 - \cos x}{\color{blue}{1 \cdot \sin x}}}\right)\]

   6. Applied *-un-lft-identity 1.0

      \[\leadsto \log \left(e^{\frac{\color{blue}{1 \cdot \left(1 - \cos x\right)}}{1 \cdot \sin x}}\right)\]

   7. Applied times-frac 1.0

      \[\leadsto \log \left(e^{\color{blue}{\frac{1}{1} \cdot \frac{1 - \cos x}{\sin x}}}\right)\]

   8. Applied exp-prod 1.1

      \[\leadsto \log \color{blue}{\left({\left(e^{\frac{1}{1}}\right)}^{\left(\frac{1 - \cos x}{\sin x}\right)}\right)}\]

   9. Applied log-pow 0.9

      \[\leadsto \color{blue}{\frac{1 - \cos x}{\sin x} \cdot \log \left(e^{\frac{1}{1}}\right)}\]

   10. Simplified 0.9

       \[\leadsto \frac{1 - \cos x}{\sin x} \cdot \color{blue}{1}\]
   11. Using strategy rm

   12. Applied div-sub 1.1

       \[\leadsto \color{blue}{\left(\frac{1}{\sin x} - \frac{\cos x}{\sin x}\right)} \cdot 1\]
   13. Using strategy rm

   14. Applied clear-num 1.2

       \[\leadsto \left(\frac{1}{\sin x} - \color{blue}{\frac{1}{\frac{\sin x}{\cos x}}}\right) \cdot 1\]

   15. Applied frac-2neg 1.2

       \[\leadsto \left(\color{blue}{\frac{-1}{-\sin x}} - \frac{1}{\frac{\sin x}{\cos x}}\right) \cdot 1\]

   16. Applied frac-sub 1.1

       \[\leadsto \color{blue}{\frac{\left(-1\right) \cdot \frac{\sin x}{\cos x} - \left(-\sin x\right) \cdot 1}{\left(-\sin x\right) \cdot \frac{\sin x}{\cos x}}} \cdot 1\]

   17. Simplified 1.1

       \[\leadsto \frac{\color{blue}{\sin x + \left(-1\right) \cdot \frac{\sin x}{\cos x}}}{\left(-\sin x\right) \cdot \frac{\sin x}{\cos x}} \cdot 1\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.014503396267940102:\\ \;\;\;\;\frac{\frac{1}{\sin x} \cdot \frac{1}{\sin x} - \frac{\cos x}{\sin x} \cdot \frac{\cos x}{\sin x}}{\frac{1}{\sin x} + \frac{\cos x}{\sin x}} \cdot 1\\ \mathbf{elif}\;x \le 0.0228016167359906367:\\ \;\;\;\;\left(\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x + \left(-1\right) \cdot \frac{\sin x}{\cos x}}{\left(-\sin x\right) \cdot \frac{\sin x}{\cos x}} \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2))

  (/ (- 1 (cos x)) (sin x)))